That is, \({P_1}\) cares directly about the contract she offers and about the aggregate of the contracts that all parties offer. Instead of thinking of \({P_1}\) as choosing only her own contract, we can think of her as choosing the aggregate contract \(w\) that will be offered to the agent. That is, write \[w = {w_1} + \sum\limits_{i = 2}^N {{w_i}} .\]

Lemma 1: If \(\bar w\) is an equilibrium aggregate contract, then \[\bar w \in \mathop {\arg \max }\limits_{w \in W\left( {\bar w} \right)} \Lambda \left( {w,\bar w} \right),\] where \[W\left( {\bar w} \right) = \bigcup\limits_{{{\bar w}_1} + \cdots + {{\bar w}_N},{{\bar w}_i} \in W} {\bigcap\limits_{i = 1}^N {W + \bar w - {{\bar w}_i}} } .\] At this point, we have shown only that if \(\bar w\) is an equilibrium aggregate contract, it solves this SGM program. That is, solving this program is a necessary, but not sufficient, condition for \(\bar w\) to be an equilibrium aggregate contract. In other words, all we have shown at this point is that if \(\bar w\) does not solve this program, it cannot be an equilibrium aggregate contract. Lemma 4 below will derive necessary and sufficient conditions for \(\bar w\) to be an equilibrium aggregate contract when Assumption S is satisfied.

I now state several lemmas that simplify this problem. Lemmas 2 and 3 simplify the set \(W\left( {\bar w} \right)\).